Quantum Electroweak Symmetry Breaking Through Loop Quadratic Contributions

Based on two postulations that (i) the Higgs boson has a large bare mass $m_H \gg m_h \simeq 125 $ GeV at the characteristic energy scale $M_c$ which defines the standard model (SM) in the ultraviolet region, and (ii) quadratic contributions of Feynman loop diagrams in quantum field theories are physically meaningful, we show that the SM electroweak symmetry breaking is induced by the quadratic contributions from loop effects. As the quadratic running of Higgs mass parameter leads to an additive renormalization, which distinguishes from the logarithmic running with a multiplicative renormalization, the symmetry breaking occurs once the sliding energy scale $\mu$ moves from $M_c$ down to a transition scale $\mu =\Lambda_{EW}$ at which the additive renormalized Higgs mass parameter $m^2_H(M_c/\mu)$ gets to change the sign. With the input of current experimental data, this symmetry breaking energy scale is found to be $\Lambda_{EW}\simeq 760$ GeV, which provides another basic energy scale for the SM besides $M_c$. Studying such a symmetry breaking mechanism could play an important role in understanding both the hierarchy problem and naturalness problem. It also provides a possible way to explore the experimental implications of the quadratic contributions as $\Lambda_{EW}$ lies within the probing reach of the LHC and the future Great Collider.

The advent of the Standard Model (SM)-like Higgs boson at the LHC [1,2] initiates us to investigate in detail the properties of the SM Higgs sector and to understand deeply the electroweak symmetry breaking mechanism. It is known that the most distinguishing feature of the SM Higgs sector is the existence of quadratic contributions or quadratic divergences in the loop calculations of Feynman diagrams. It involves the long-lasting issues on the so-called hierarchy problem and naturalness problem. The main questions include whether the Higgs boson is a fundamental particle or a composite one, and what is the energy scale for a possible new physics beyond the SM. As it is well understood that the gauge couplings of interactions and masses of particles are all running quantities of the energy scale. The well-known fact is the discovery of asymptotic freedom of gauge coupling in QCD [3,4], which has led to the motivation of grand unification theory for all gauge interactions. The running behavior of all physical quantities in the standard model is logarithmic except for the Higgs mass parameter. For the logarithmic running, it is known to result in a multiplying renormalization. As a consequence, such a running is always proportional to the coupling/mass itself. For the Higgs mass parameter, if ignoring the quadratic contributions, one yields a similar behavior. When taking into account the quadratic contributions to the Higgs mass parameter, one yields a quadratic running for the Higgs mass parameter. Unlike the logarithmic running, the quadratic running will result in an additive renormalization. In fact, it was shown that the quantum gravitational contributions can also cause a quadratic running to gauge couplings [5,6]. How to treat and understand the quadratic running of the Higgs mass parameter comes to the main issue in this paper.
It has been known for a long time that quantum field theory (QFT) is plagued by ultraviolet (UV) divergences. In the modern viewpoint based on the Wilsonian effective field theories [7,8], QFT is typically defined with respect to some physical UV cutoffs, and the UV divergences are reinterpreted as extra contributions from the UV modes to the parameters in the low-energy effective Lagrangians. In this sense, we shall use the words "UV contribution" rather than "UV divergence" in the quantum loop calculations. In the Higgs sector of the SM, there is a long-established implication that the quadratic contributions, unlike the logarithmic contributions, can lead to unwanted over-contributions from the UV modes to the low-energy Higgs mass parameter, unless there exist some extremely delicate tunings between these quadratic contributions and the so-called bare Higgs mass parameter. Therefore, to achieve a natural explanation for the observed Higgs mass rather than asking the huge fine-tunings, it is important to treat properly these quadratic contributions. This is often called the naturalness problem in literature. Meanwhile, if there is no new physics   above the electroweak scale up to the grand unification scale or Planck scale, it then raises the so-called hierarchy problem. In the last three decades, these problems have drawn enormous attentions. To solve these problems, one chooses either to eliminate these quadratic contributions in the UV regions or to lower the ultimate scale from the Planck scale to the TeV scale. Another solution is based on the consideration that the discovered Higgs is not a fundamental scalar particle, so that the calculations for the quadratic contributions cannot be extended to scales much higher than the characteristic energy scale of the composite Higgs models. Alternatively, one even argues that the naturalness problem is not a right question at all, and nature just behaves in an unnatural way. Popular studies along these directions include the electroweak scale supersymmetry [9], extra dimension models [10], composite Higgs models [11], multiverse scenario [12]. Notably, almost all the solutions (with the multiverse scenario as a representative counterexample) can lead to new physics at the TeV scale. In all the considerations, the quadratic contributions are treated to be the inevitable loop quantum effects in QFTs and have to be tamed carefully for avoiding the intolerable unnaturalness.
On the other hand, it has been demonstrated that in the effective field theories, such as the chiral dynamical model of QCD, the quadratic contributions play a crucial role for the derivation of the gap equation to describe the dynamically generated spontaneous chiral symmetry breaking [13,14], where the scalar mesons have been regarded as the composite Higgs-type bosons [14]. In the chiral perturbation theory, the quadratic contributions have been shown to be significant for understanding the ∆I = 1/2 selection rule on the isospin amplitudes A 0 /A 2 [15] and the direct CP-violating parameter / [16], and to provide simultaneously a consistent explanation for both the ratio / and the isospin amplitudes A 0 /A 2 in the Kaon decays [17]. Recently, it was noticed in Ref. [18] that the coefficient of the quadratic contributions of the SM Higgs sector has a novel zero point around the scale of 10 23 GeV, which provides new insights into the Higgs inflation scenario [19,20] and also the possible hierarchy problem solution [21] based on the idea of softly broken conformal symmetry [22].
Theoretically, in the perturbative expansions and calculations of the SM, there exist in general both quadratic and logarithmic divergences. When adopting the dimensional regularization scheme, the quadratic divergences are suppressed due to the analytical extension for the space-time dimensions of original theories. Although the dimensional regularization scheme is practically very convenient in calculations and widely recognized in literature, it is inevitable to result in divergences when taking the exact space-time dimensions to recover the original theories. Thus the dimensional regularization scheme is actually in spirit incompatible with the modern framework of effective field theories, especially when the theories involve quadratic contributions. It is natural to find out a new scheme which is suitable for the modern framework of effective field theories and applicable for all QFTs with preserving symmetry properties and divergent structures of original theories. It has turned out that the loop regularization (LORE) method proposed in Ref. [23] is a concrete realization for such new schemes. It has been shown explicitly at one-loop level that the LORE method can preserve Poincare symmetries and gauge symmetries [23], even supersymmetry [24], and can be extended consistently beyond the one loop [25,26]. Unlike the dimensional regularization scheme, the LORE method is realized in the exact space-time dimensions of original theories and all the calculations can be done exactly without modifying original theories. All the UV contributions, both quadratic and logarithmic, in the Feynman integrals can be calculated in a unified manner as the divergent integrals can well be defined in the LORE method.
In particular, the LORE method is found to be an infinite-free regularization scheme which leads to finite results characterized by two intrinsic energy scales. These two intrinsic energy scales are introduced naturally in the LORE method to play the roles as the characteristic energy scale M c and the sliding energy scale µ, which can be identified as the UV energy scale to define the so-called bare Lagrangians and the infrared (IR) cutoff to yield the low energy effective Lagrangians respectively. As a consequence, the LORE method enables us to define a finite renormalization theory of QFTs. For more details on the LORE method, it is referred to the original papers and the recent review [27] for the interested readers.
where m H , λ H and y t are the bare Higgs boson mass, Higgs coupling constant and top-quark Yukawa coupling constant respectively, they all are defined at the UV characteristic energy where g 2 and g 1 are the gauge couplings for the SU (2) L and U (1) Y group respectively at the UV characteristic energy scale M c . I a W is related to the Pauli matrices via I a W = σ a /2. Other interaction terms are the same as the ones in the SM with all the couplings defined at the UV characteristic energy scale M c .
In this article we will pay attention to the issue on the quadratic contributions to the above Higgs sector. At one loop level, we perform a calculation by using the loop regularization method and obtain a finite renormalized result for the one loop contributions to the Higgs mass parameter. The additive renormalized Higgs mass parameter at a truncated energy scale µ is explicitly given by are given by: for the Higgs mass parameter, and for the coupling constants. They hold for the sliding energy scale µ truncated above the heavy top quark mass µ > m t . Note that the strong interaction has no direct interactions to the Higgs field, but it does enter on the stage in the RGE approach through the g 3 dependence in the beta function of the top Yukawa coupling y t .
It is seen from the above RGE Eq.(3) that the quadratic running leads to an additive renormalization, which is different from the logarithmic running that gives a multiplying renormalization. Taking into account the above evolutions of the Higgs mass parameter and coupling constants as the energy scale µ, we are going to demonstrate the implications of the quadratic contributions in the Higgs sector. For our present purpose, it is more convenient to take the integrated expression for the additive renormalized Higgs mass parameter m 2 H (M c /µ) defined in Eq. (2) to show the properties of the loop quadratic contributions. For a numerical calculation, we will take the RGE approach to obtain more accurate quantitive results via Eq. (3). It is interesting to observe that when the coefficient associated to the quadratic contributions is positive, i.e., there must exist a phase transition point µ = Λ EW at which the additive renormalized Higgs mass parameter will approach to vanish for a proper UV characteristic energy scale which shows that the large bare Higgs mass at the characteristic energy scale M c gets to become smaller and vanishing at a low energy phase transition point Λ EW through quadratic running.
It becomes manifest that below such a phase transition point, i.e., µ < Λ EW , the additive renormalized Higgs mass parameter changes the sign. As a consequence, the Higgs potential gets unstable and the electroweak symmetry will be broken down spontaneously. To be more explicit, we express the Higgs sector defined below the IR cutoff scale Λ EW as follows with the definitions where Λ EW is regarded as the electroweak symmetry breaking/restoration energy scale.
When the electroweak symmetry gets broken down, the CP even neutral component of the Higgs doublet receives a nonzero evolving vacuum expectation value (eVEV), which is parametrized as the following form where the Higgs field h(x) corresponds to the quantized physical degree of freedom observed at the LHC. v h is the eVEV given by Such a symmetry breaking may be referred simply as a quantum electroweak symmetry breaking (QEWSB) as it is induced by the quadratic contributions of loop quantum effects.
It is seen that the quantum loop quadratic contribution of top quark is crucial for the QEWSB as it mainly causes the additive renormalized Higgs mass parameter changing the sign. When the sliding energy scale firstly approaches to the top quark mass µ ∼ m t as the top quark is the heaviest particle in the SM, which enables us to estimate the QEWSB scale Λ EW Here m t is regarded as the direct measurement top mass m t = 173.34 GeV. In the numerical calculations, we shall take the RGEs (3) For the boundary conditions at the top mass m t = 173.34 GeV, we take the latest extracted results presented in Ref. [30] g 1 (µ = m t ) = 0.35830, g 2 (µ = m t ) = 0.64779, g 3 (µ = m t ) = 1.1666 , Here we only take the central values as their errors without impacting on our final conclusion. Taking the above input as the boundary conditions for the RGEs (3)-(6), we find the electroweak symmetry restoration scale Λ EW to be Λ EW 684 GeV. Here only the oneloop RGEs have been adopted, and also only the quadratic contributions have been taken into account for the additive renormalized Higgs mass parameter m 2 H (M c /µ). To estimate the effects from higher loop contributions, we have checked by using the RGEs presented in Ref. [30] that the impact from the higher loop contributions and logarithmic contributions to m 2 H (M c /µ) is negligibly small, which increases Λ EW only by about 1%, namely Λ EW ∼ 700 GeV. The additive renormalized Higgs mass parameter m 2 H (M c /µ) is shown in Fig. 1: When the sliding energy scale passes through the top mass threshold, i.e., µ ≤ m t , the top quark will decouple effectively from the theory according to the usual assumption of effective field theory. This decoupling effect leaves a pattern on the later-on additive renormalized Higgs mass parameter as follows for µ ≤ m t (15) which shows that the decoupling of the top quark leads the Higgs mass parameter µ 2 h to be decreased or m 2 H (M c /µ) to be less negative. As a consequence, the eVEV v 2 h (Λ EW /µ) gets a bit smaller. While such a decrease only happens in a small energy range as the sliding scale immediately moves down to the Higgs mass threshold µ = m h = 125.15 GeV [30]. When µ goes down further to be less than the Higgs mass m h , the Higgs boson itself also decouples in much the same way as the top quark, which eventually freezes the renormalization of the Higgs mass parameter µ 2

The actual change of the eVEV ∆v
The numerical value of the eVEV v h (Λ EW /µ) gets fixed and remains the same down to The property of the additive renormalized Higgs mass parameter is illustrated in Fig. 2, where the anomalous changing between µ = m h and µ = m t is exaggerated.
For the logarithmic running with multiplying renormalization, the resulting Lagrangian can be shown to be scale-independence due to the cancellations of µ dependence among the coupling/mass renormalization and quantum fields renormalizations. Note that in applying for the renormalization scheme, a basic cut-off energy scale or a cut-off energy scale at a typical mass of particle is implicitly introduced to keep full control of the scale-independence.
For µ = M c Λ EW ∼ 700 GeV, the bare Higgs mass is approximately given by where the UV characteristic energy scale can in principle be taken to be the Planck scale Let us now make another interesting issue. That is whether the QEWSB mechanism in the SM is consistent with the expanding universe for µ ≤ M c . More concretely, it is important to ensure that the Higgs boson with a renormalized mass is stable during the thermal history of our universe. It is widely believed that after the reheating the universe enters into the radiation dominant phase and starts the hot big bang. Then, the temperature T of the cosmic plasma provides an effective measure of the typical energy scale for the particle physics processes taking place at that time. For T > Λ EW , the SM lives in the electroweak symmetry phase and all the SM particles except the Higgs boson are massless.
It is crucial to make sure that the Higgs boson does not decouple from the cosmic plasma too early, even before the QEWSB. Otherwise, the QEWSB mechanism does not work and all the SM particles cannot gain their mass from the electroweak symmetry breaking. The SM Higgs boson couples to the top quark most strongly and the coupling constant y t is roughly order one. For T > Λ EW , the Higgs boson can decay into two massless top quarks and vice versa. The rate of this interaction is roughly Γ ∼ T . Given the Hubble rate H ∼ T 2 /M P l , we have Γ/H ∼ M P l /T > 1 , for Λ EW < T < M P l (21) which shows that the SM Higgs boson can always stay equilibrium with the cosmic plasma before the QEWSB.
In conclusion, we have demonstrated the QEWSB mechanism in the SM based on two basic postulations that the Higgs boson has a bare mass at the UV characteristic energy scale Considering the fact that the energy scale of the hard scattering processes at the LHC has already reached the TeV scale, which is greater than the electroweak symmetry restoration scale Λ EW ∼ 700 GeV predicted based on the QEWSB mechanism, it should be interesting to probe the possible physics effects around the electroweak symmetry breaking/restoration scale Λ EW ∼ 700 GeV and test the QEWSB mechanism at the LHC and future Great Collider, which will be significantly important for understanding both the hierarchy problem and naturalness problem.